Some part of mesh won't deform
The automatic weight feature in blender isn't perfect, pretty much every time you use it you will have to go back and manually perfect some weights.

How to fix it:


Hop into to weight paint mode (Click on the UI button that currently says either object or edit mode in the 3D Viewport, and select the Weight Paint Feature)
Open up the current object info tab (The box tab on the same 'tab group' as the render tab) and click on the index finger vertex group.
Use the weight paint brush to change the the effect the bone has on the mesh (Red is full parent and blue is no effect)


Happy Blending!